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ckrao.wordpress.com | ||
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dominiczypen.wordpress.com
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| | | | | For $latex A, B \subseteq \omega$ we write $latex A \subseteq^* B$ if $latex A\setminus B$ is finite, and we write $latex A\simeq^* B$ if $latex A\subseteq^* B$ and $latex B\subseteq^* A$. A tower is a collection $latex {\cal T}$ of co-infinite subsets of $latex \omega$ such that for all $latex A\neq B\in {\cal T}$... | |
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algorithmsoup.wordpress.com
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| | | | | The ``probabilistic method'' is the art of applying probabilistic thinking to non-probabilistic problems. Applications of the probabilistic method often feel like magic. Here is my favorite example: Theorem (Erdös, 1965). Call a set $latex {X}&fg=000000$ sum-free if for all $latex {a, b \in X}&fg=000000$, we have $latex {a + b \not\in X}&fg=000000$. For any finite... | |
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mikespivey.wordpress.com
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| | | | | It's fairly well-known, to those who know it, that $latex \displaystyle \left(\sum_{k=1}^n k \right)^2 = \frac{n^2(n+1)^2}{4} = \sum_{k=1}^n k^3 $. In other words, the square of the sum of the first n positive integers equals the sum of the cubes of the first n positive integers. It's probably less well-known that a similar relationship holds... | |
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www.jeremykun.com
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| | | Last time we investigated the naive (which I'll henceforth call "classical") notion of the Fourier transform and its inverse. While the development wasn't quite rigorous, we nevertheless discovered elegant formulas and interesting properties that proved useful in at least solving differential equations. Of course, we wouldn't be following this trail of mathematics if it didn't result in some worthwhile applications to programming. While we'll get there eventually, this primer will take us deeper down the rabbit hole of abstraction. | ||
